Methods and apparatus for analyzing transmission lines with decoupling of connectors and other circuit elements

ABSTRACT

Methods and apparatus are provided for analyzing transmission lines with decoupling of connectors and other circuit elements. According to one aspect of the invention, circuits with one or more parasitic elements are analyzed by partitioning at least one of the parasitic elements in a transverse manner; identifying a plurality of subcircuits each comprised of partitioned circuit elements from the plurality of transmission lines and one or more parasitic elements in a given path; wherein each of the subcircuits is associated with a path in the circuit; performing a waveform relaxation analysis between each of the subcircuits; and repeating the step of performing the waveform relaxation analysis using waveforms determined in a previous iteration until convergence to a resultant waveform has occurred. The circuit can optionally further comprise one or more transmission lines which would also be partitioned in a transverse manner.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/356,636, filed Feb. 17, 2006, now abandoned, which is incorporatedherein by reference.

FIELD OF THE INVENTION

The present invention relates to the analysis of electrical circuits,and more particularly, to the analysis of transmission line circuitsusing Transversal Waveform Relaxation techniques.

BACKGROUND OF THE INVENTION

Electrical circuits with transmission lines are typically analyzed toensure proper functioning of the circuits. The coupling between multiplelines and the resultant coupled signals are an important aspect of thesetransmission line circuits. Power distribution systems, for example,often must be analyzed for stability and other properties. Similarly, ininstrumentation circuits and computer circuitry in tacks or cabinets thenoise coupled between transmission lines needs to be understood andminimized.

A number of techniques have been proposed or suggested fob analyzingmultiple wire transmission lines. Such techniques are described, forexample, in Clayton Paul, Analysis of Multiconductor Transmission Lines,Ch. 5 (Wiley, 1994). While these techniques are suitable for theanalysis of models with a few lines, the complexity increases rapidly asthe number of lines increases. Some simplified techniques have beenproposed to approximate the solution for many transmission lines withonly neighbor-to-neighbor wire coupling. These approaches are suitablewhere reduced accuracy is acceptable to gain speed.

Existing techniques for analyzing multiple wire transmission lines arelimited in the number of coupled lines or wires that can be analyzedsimultaneously. The complexity of the coupling calculation increasesrapidly as the number of lines increases, and the accuracy of theresults decreases with the increasing number of lines. Hence, theexisting techniques are unable to handle a large number of lines due toexcessive computation time and the results become questionable. Someprior art techniques ignore the couplings for more than two lines tospeed up the process. Other techniques are based on having only linearcircuits to speed up the calculation process and are thereforeunsuitable for handling even typical transmission line circuits, whichinclude surrounding nonlinear drivers and receivers.

U.S. patent application Ser. No. 10/776,716, entitled “System and MethodFor Efficient Analysis of Transmission Lines,” incorporated by referenceherein, discloses “Transversal Waveform Relaxation” techniques foranalyzing multiple wire transmission lines by determining which sourcesinfluence each of a plurality of transmission lines, based on couplingfactors. Transmission line parameters are computed based on the sources,which influence each transmission line. A transient or frequencyresponse is analyzed for each transmission line by segmenting each lineto perform an analysis on that line. The step of analyzing is repeatedusing waveforms determined in a previous iteration until convergence toa resultant waveform has occurred. For a more detailed discussion ofsuch Transversal Waveform Relaxation techniques, see, for example,Nakhla et al., “Simulation of Coupled Interconnects Using WaveformRelaxation and Transverse Partitioning,” EPEP'04, Vol 13, pp 25-28,Portland, Oreg., October 2004, incorporated by reference herein.

While such Transversal Waveform Relaxation techniques have greatlyimproved the analysis of multiple wile transmission lines and provide anefficient approach for transverse decoupling of transmission lines, theysuffer from a number of limitations, which if overcome, could providefurther improvements. In particular, the connectors in such transmissionlines are still coupled. A need therefore exists for a TransversalWaveform Relaxation algorithm that decouples entire paths from eachother.

SUMMARY OF THE INVENTION

Generally, methods and apparatus are provided for analyzing transmissionlines with decoupling of connectors and other circuit elements.According to one aspect of the invention, circuits with one or moreparasitic elements are analyzed by partitioning at least one of theparasitic elements in a transverse manner; identifying a plurality ofsubcircuits each comprised of partitioned circuit elements from theplurality of transmission lines and one or more parasitic elements in agiven path; wherein each of the subcircuits is associated with a path inthe circuit; performing a waveform relaxation analysis between each ofthe subcircuits; and repeating the step of performing the waveformrelaxation analysis using waveforms determined in a previous iterationuntil convergence to a resultant waveform has occurred. The circuit canoptionally further comprise one or mole transmission lines which wouldalso be partitioned in a transverse manner.

Each of the circuit elements, such as the transmission lines andparasitic elements, are represented using a circuit macromodel. Thecircuit macromodel separates each path that enters and exits a givenelement such that it is coupled only with elements suitable for waveformrelaxation analysis.

According to another aspect of the invention, electrical couplingsbetween the paths of the circuit are embodied using current controlledsources. At each step of the waveform relaxation analysis, the states ofthe sources are updated using the path states from a previous step.

A more complete understanding of the present invention, as well asfurther features and advantages of the present invention, will beobtained by reference to the following detailed description anddrawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram illustrating a system in accordance with anembodiment of the present invention;

FIG. 2 is a flow diagram illustrating a method for transmission lineanalysis using a transverse waveform relaxation process;

FIG. 3 depicts an illustrative geometry for a multiple transmissionlines to be analyzed in accordance with the present invention;

FIG. 4 illustrates an exemplary transmission line system that can beanalyzed using the disclosed techniques;

FIG. 5 illustrates a flow chart describing an exemplary TransversalWaveform Relaxation process incorporating features of the presentinvention;

FIG. 6 illustrates an exemplary four-port circuit having twoillustrative poles;

FIG. 7 illustrates the partitioning for the circuit of FIG. 6;

FIG. 8 illustrates an exemplary non-physical equivalent circuit that canbe made for a number of state variables;

FIG. 9 illustrates exemplary macromodel circuits for the partitionedfoul terminal circuit of FIG. 7;

FIG. 10 is a flow chart describing an exemplary implementation of acircuit preprocessor;

FIG. 11 is a flow chart describing an exemplary implementation of aconnector partitioning process; and

FIG. 12 is a flow chart describing an exemplary implementation of aTransversal Waveform Relaxation process according to the presentinvention

DETAILED DESCRIPTION

The present invention provides methods and apparatus for analyzingtransmission lines with decoupling of connectors, such that entire pathsare decoupled from each other. The electrical couplings between thepaths of the multipath circuit are embodied using current controlledsources. At each step of the Transversal Waveform Relaxation process,the states of these sources are updated using the path states from theprevious step. Since the capacitive and inductive couplings aregenerally weak in a multipath design, the Transversal WaveformRelaxation process converges in a rapid manner to the correct signalwaveforms for all paths.

First, a discussion is presented of suitable techniques for TransversalWaveform Relaxation. Thereafter, a detailed discussion is provided ofvarious aspects of the present invention, in a section entitled“Decoupling of Connectors.”

Transversal Waveform Relaxation Background

FIG. 1 is a block diagram illustrating a system 100 in which the presentinvention can operate. As shown in FIG. 1, an exemplary system 10includes a computer 12, such as a personal computer or a mainframe.Computer 12 includes any interface devices known in the art. Computer 12may include a plurality of modules or software packages that may beresident in the system or coupled thereto via a network or the like. Forexample, computer 12 may be provided access to electronic designautomation (EDA) libraries or other circuit databases 14, which includeelectrical circuits or integrated circuit chip designs.

A module 16 may include one more programs or subroutines for carryingout methods in accordance with the present invention. Module 16 mayinclude transmission line analysis programs, including a solver 17 orcode to determine coupling factors, perform sliding calculations, updatecoupling models and perform transient analysis, among other things aswill be described in greater detail herein below. Module 16 may beincorporated into other programming packages, such as lull-blown circuitanalysis systems or programs. In addition, as discussed further below inconjunction with FIGS. 5 and 10-12, the module 16 includes one or moreprocesses 500, 1000, 1100 and 1200 for implementing the presentinvention.

A computer aided design (CAD) module or program 18 may be included toimport designs or design information to the system 10 to provide theappropriate circuit analysis CAD schematics and or EDA data fromdatabase 14 may be employed as inputs to module 16 to analyze componentsof a design, and preferably transmission lines in the design.

FIG. 2 is a flow diagram illustrating a method 200 for transmission lineanalysis using a transverse waveform relaxation process. Thetransmission line analysis process 200 may be referred to as atransverse waveform relaxation process. In circuit designs, one or moretransmission lines may be present. To handle a plurality of coupledlines, the impact of each neighboring transmission line needs to beconsidered. For a more detailed discussion of suitable transversewaveform relaxation processes, see, for example, U.S. patent applicationSer. No. 10/776,716, entitled “System and Method For Efficient Analysisof Transmission Lines,” or Nakhla et al, “Simulation of CoupledInterconnects Using Waveform Relaxation and Transverse Partitioning,”EPEP'04, Vol 13, pp 25-28, Portland, Oreg., October 2004, eachincorporated by reference herein.

Generally, the transversal waveform relaxation algorithm 200 initializesthe electrical states of all wiles; analyzes every wire at a time withcouplings from all other wires implemented with known voltage andcurrent sources; and iterates until convergence.

The exemplary transverse waveform relaxation process 200 shown in FIG. 2addresses neighbor-to-neighbor coupling early on in the process. Thiskeeps the problem sparse if the model includes a large number of coupledtransmission lines. Also, a coupling model of parameters, which have tobe computed at the same time, is limited to a relatively small number.The model limits the number of coupled sources needed to represent thecouplings. Then, each wile is individually analyzed as a subcircuitwhile the coupled sources are taken into account, and the waveformsbetween the last two iterations are compared to check for convergence.

In block 202, coupling factors are determined for transmission lines ina given design Coupling factors are determined by calculating theinfluence of neighboring lines on a given line. In one embodiment, aninductance coupling factor (cf) may be calculated as:

${cf} = \frac{L_{12}}{\sqrt{L_{11}L_{22}}}$where L₁₂ is the inductive influence of L₂ on L₁ (coupling term) and L₁₁is the inductance of line 1 and L₂₂ is the inductance of line 2 (thesemay be referred to as self terms). A similar coupling factor can becalculated for resistance

$\left( {{cf} = \frac{R_{12}}{\sqrt{R_{11}R_{22}}}} \right)$and capacitance

$\left( {{cf} = \frac{C_{12}}{\sqrt{C_{11}C_{22}}}} \right).$

Other coupling factors may be used and preferably dimensionlessvariables. Also, default values, e.g., 0001, may be inserted into theprogram for coupling factors to ensure a nonzero number exists in thecase where the coupling factor is not available or for other reasons.

The coupling factors are estimated first with approximate computations.This leads to the knowledge of how many transmission lines are to beincluded in each segment of the calculation. The segments are computedin an overlapping way such that all the interactions are taken intoaccount so that each line can be analyzed individually taking intoaccount the pertinent couplings for each line.

After calculating the coupling factors for transmission lines in adesign, the coupling factors are compared to a threshold value(s), inblock 204, to determine if they will have an influence on neighboringlines. For example, L_(coupling)<L_(tolerance);C_(coupling)<C_(tolerance); and R_(coupling)<R_(tolerance).

The tolerance or threshold is preferably set by a designer but can alsobe calculated based on parameters or criteria for a given design. Forexample, in sensitive equipment, a smaller tolerance may be neededmeaning smaller influences should be considered in analyzingtransmission line parameters. Coupling factors that are determined to betoo small may be disregarded in future calculations for a given segment.However, since many circuits are dynamic and different portions of acircuit may be operational at a different time, different time framesmay be investigated to ensure a complete solution.

For example, for a given line, coupling factors are employed todetermine the influence of other lines on the line in question. Based onthese estimates, the calculations are segmented for each line to includethe most influential coupling effects.

In block 206, a sliding computation is performed to calculatetransmission line parameters (L, C and R). These calculations arepreferably based on geometric features. The calculation of the per-unitline parameters L, R, and C is preferably performed in a segmented way,since the simultaneous calculation of these matrix quantities is veryexpensive for more than a few lines. Hence, each segmented calculationwill include the computation of the L, R, C parameters for a number oflines. The number of steps needed to determine these parameters is muchsmaller than the total number of lines since the most influential linesare considered. This simplifies the evaluation of the L, R, C parameterssince each sub-problem is much smaller than the large single evaluationof each parameter.

For example, assuming 100 transmission lines, the coupling evaluationdetermines that 5 lines should be included neat each line to accuratelytake the coupling into account. So for the sliding calculation, thefirst 15 lines are evaluated simultaneously using, for example, astandard field solver for L, R and C. This result can be used for thefirst 10 lines in the transverse waveform relaxation (WR) approach givenherein. Then, in a next sliding field calculation, the next L, R, C, areevaluated for 15 lines from line 10 to line 25. Then, this calculationcan be used to evaluate the transverse WR for lines 11 to 20, and so on.Hence, the field calculation is completed only on a subset of the lineswhich is much faster since the compute time of the field solverincreases enormously with the number of lines considered.

In one embodiment, the characteristics of a circuit as defined in a CADschematic are employed to make these sliding calculations. The slidingcalculations provide a baseline for the transient analysis as will bedescribed hereinafter.

Based on the sliding computation, in block 208, a coupling model ormodels are employed to reduce the circuit characteristics into terms ofvoltage and/or current sources with lumped elements (L, R, C) oralternately uses the method of characteristic models to model thecircuit.

In block 210, a transient of frequency domain analysis of thetransmission lines is performed preferably one wire at a time. Thetransient/frequency analysis is based on a transmission line response tosurrounding circuits using coupled sources to other coupled lines asprovided by the models set forth in block 208.

In one embodiment, partitioning along the coupling of the lines isperformed. In other words, each line is taken one at a time consideringthe most pertinent coupling influences on that line. Alternately,partitioning over the length of the line may be performed as well or inaddition to a calculation for the partitioning of the coupling of theline.

In block 212, the transient analysis of block 210 is repeated untilconvergence is achieved by comparing a previous value of the waveformsdetermined by the transient analysis from a previous iteration to thewaveforms determined in the present iteration. If convergence isachieved the resultant waveforms have been determined and are availablein block 214. If convergence has not yet been achieved, then the programreturns to block 210 to recalculate the waveforms.

As previously indicated, the present invention provides methodsdiscussed further below in conjunction with FIG. 4 to determine thenumber of iterations needed for a computed solution to achieve a givenlevel of accuracy.

FIG. 3 depicts an illustrative geometry 300 for a multiple transmissionlines to be analyzed in accordance with the present invention.Transmission lines 302 are numbered 1 to N in the depicted section of acircuit 300. Lines 302 are marked with an A to indicate that they areaggressor lines. These lines are exited with some external circuitry. Incontrast, the lines which are marked with a V are victim lines which arenot excited with external sources.

Using one method, e.g., set forth with reference to FIG. 2, thesubcircuits/lines of FIG. 3 are analyzed starting at line 1, insequence, until line N is reached. This sequence is followed for eachcalculation in FIG. 2. Then, the sequence is repeatedly followed gountil convergence in the transient/frequency analysis (e.g., blocks 210and 212).

A more efficient method is based on signal flow. Fox example, initially,all coupled waveform sources are set to zero. Then, starting with theanalysis of the circuits, which include the aggressors (A) first, newcoupled-source quantities are available from the coupling model (block206). Then, the nearest neighbors are analyzed since they will includethe largest signals next to the aggressors. The process progressesthrough all the wires until all of the wires have been visited. In eachstep/iteration, the latest, updated waveforms are employed.

These methods are directly applicable to parallel processing for circuitproblems, which include transmission lines. Each of the N transmissionlines forms a separate subsystem with a transverse decoupling scheme(e.g., portioning along coupling lines or effects). Further partitioningis possible along the length of the line using conventional techniques.

Partitioning or segmenting line by line (coupling) leads to 2Nsubsystems which can be analyzed on separate processors where the onlyinformation that needs to be exchanged between processors is waveforms.Hence, an enormous gain in speeding up the process by parallelprocessing is achieved.

Decoupling of Connectors

As previously indicated, the present invention provides methods andapparatus for analyzing transmission lines with decoupling ofconnectors, such that entire paths are decoupled from each other. Theelectrical couplings between the paths of the multipath circuit areembodied using current controlled sources. At each step of theTransversal Waveform Relaxation process, the states of these sources areupdated using the path states from the previous step. Since thecapacitive and inductive couplings are generally weak in a multipathdesign, the Transversal Waveform Relaxation process converges in a rapidmanner to the correct signal waveforms for all paths.

FIG. 4 illustrates an exemplary transmission line system 400 that can beanalyzed using the disclosed techniques. As shown in FIG. 4, a pluralityof drivers 410-1 through 410-N drive a plurality of paths 415, one ormore of which may be connected by one or more connectors 420-1 through420-M (hereinafter, collectively referred to as connectors 420), througha number of transmission lines 430. The transmitted signals are receivedby one or more receivers 450-1 through 450-N.

The transmission lines 43-0 can be analyzed using the TransversalWaveform Relaxation techniques referenced above. The present inventionprovides a Transversal Waveform Relaxation algorithm that decouples theconnectors 420 as well.

In general, the techniques of the present invention can be applied toany system, such as the system 400, having basic circuit elements, suchas drivers 410, receivers 450, connectors 420 and transmission lines430, as well as other potential coupled parasitic circuits (not shown).According to one aspect of the invention, each circuit model for eachbasic circuit element must be decoupled such that each path can beanalyzed independently. The circuits must be designed such that thecoupling factors between each of the paths is small, typically k<<0.2,in a known manner. It has been shown that for such small coupling cases,the Transversal Waveform Relaxation approach converges in a rapidmanner.

The coupling factor between paths can approximately be defined for thelarge class of inductive k₁ and/or capacitively coupled circuits k_(c)ask ₁ =L ₁₂ /L ₂₂  (1)andk _(c) =C ₁₂ /C ₂₂  (2)Here, C₁₂ and L₁₂ are the coupling factors in the models for thedifferent elements and C₂₂ and L₂₂ are the self elements into which thecoupling occurs. These path-to-path coupling factors imply that thecoupling, while being very important for the accurate analysis of theproblem, are moderate. These considerations apply to the generalsituation even if they are generally applicable.

As discussed further below, each of the model elements in the generalcircuit 400 given in FIG. 4 is replaced with a circuit model that allowseach individual path to be separated within the model with a WaveformRelaxation coupling only. A formulation is provided fox a macromodelthat is applicable to a wide set of different structures. In addition,as discussed further below in conjunction with FIG. 5, the componentparts from each path through the system 400 form a subcircuit, such asthe subcircuit SCkt 2 shown in FIG. 4

FIG. 5 illustrates a flow chart describing an exemplary TransversalWaveform Relaxation process 500 incorporating features of the presentinvention. As shown in FIG. 5, the Transversal Waveform Relaxationprocess 500 initially represents each model element in the equivalentcircuit during step 510 by a circuit macromodel, as discussed furtherbelow in conjunction with FIGS. 6-9. The circuit macromodels separateeach path that enters and exits a given element such that it is coupledonly with elements suitable for waveform relaxation analysis. Foxexample, the model for connector 420-1 in FIG. 4 should be divided intoat least three macromodels, one fox each through connection.

As discussed further below in conjunction with FIG. 11, a subcircuitSCkt is formed during step 520 from the component parts from each paththrough the system 400. For example, a first subcircuit, SCkt 1,includes Driver 1 510-1, Part 1 of Connector 1 520-1, the first line inthe multi-transmission line 430, the first part of Connector 3 420-3 andfinally Receiver 1 450-1

A waveform relaxation (WR) analysis, such as those referenced above, isperformed during step 530 between each of the subcircuits, SCkts. Theresults are updated with new results as the resultant waveforms becomeavailable for each of the subcircuits, SCkts.

Finally, the Transversal Waveform Relaxation process 500 iterates duringstep 540 according to a schedule, such as a sequential schedule of SCkt1, SCkt 2, . . . , SCkt N. If the coupling is weak enough, then thisschedule needs to be executed only a few times. Also, for weakcouplings, integration can be done over all time rather than in timewindows.

Partitioned Macromodels

A multiport circuit can be characterized in terms of different butequivalent parameters, such as Y (admittance), Z (impedance), H(hybrid), or S (scattering). A formulation based on the admittance Yparameters is considered below for such a network as

$\begin{matrix}{{\begin{bmatrix}Y_{11} & Y_{12} & \ldots & Y_{1\; n} \\Y_{21} & Y_{22} & \ldots & Y_{2\; n} \\\ldots & \ldots & \ldots & \ldots \\Y_{n\; 1} & Y_{n\; 2} & \ldots & Y_{nn}\end{bmatrix}\mspace{14mu}\begin{bmatrix}{V_{1}(s)} \\{V_{2}(s)} \\\ldots \\{V_{n}(s)}\end{bmatrix}} = \begin{bmatrix}{I_{1}(s)} \\{I_{2}(s)} \\\ldots \\{I_{n}(s)}\end{bmatrix}} & (3)\end{matrix}$Since each entry Y_(ij) is expressed in the complex frequency, thestandard approach is to invoke some type of least squareapproximation-based technique to fit the data from sample frequencypoints to a complex rational function H(s), where

$\begin{matrix}{{H(s)} = {g + {\sum\limits_{i = 1}^{q}\frac{r_{i}}{s - p_{i}}}}} & (4)\end{matrix}$where p_(i) and r_(i) are the i-th pole-residue pair, q is the totalnumber system poles and g is quotient.

FIG. 6 illustrates an exemplary four-port circuit 600 having, forillustrative purpose, two poles. The exemplary circuit 600 is thesmallest case that includes sufficient complexity while illustrating theimportant aspects of such a model. The transfer function for thiscircuit 600 can be described by

$\begin{matrix}{\begin{bmatrix}{g_{11} + {\sum\limits_{i = 1}^{2}\frac{r_{i}^{11}}{s - p_{i}}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{12}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{13}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{14}}{s - p_{i}}} \\{\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{21}}{s - p_{i}}} & {g_{22 +}{\sum\limits_{i = 1}^{2}\mspace{11mu}\frac{r_{i}^{22}}{s - p_{i}}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{23}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{24}}{s - p_{i}}} \\{\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{31}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{32}}{s - p_{i}}} & {g_{33} + {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{33}}{s - p_{i}}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{34}}{s - p_{i}}} \\{\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{41}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{42}}{s - p_{i}}} & {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{43}}{s - p_{i}}} & {c_{44} + {\sum\limits_{i = 1}^{2}\;\frac{r_{i}^{44}}{s - p_{i}}}}\end{bmatrix}{\quad{\begin{bmatrix}{V_{1}(s)} \\{V_{2}(s)} \\{V_{3}(s)} \\{V_{4}(s)}\end{bmatrix} - \begin{bmatrix}{I_{1}(s)} \\{I_{2}(s)} \\{I_{3}(s)} \\{I_{4}(s)}\end{bmatrix}}}} & \lbrack 5)\end{matrix}$

It can be observed that each element in the above matrix of equation (5)consists of several parts representing the various mechanisms for thepaths and the couplings. It should be noted that the formulation allowsa resistive coupling among all the elements. However, the throughconnection wire resistance is much smaller than any other couplingmechanism, as discussed above FIG. 7 illustrates the partitioning 700for the circuit 600 of FIG. 6. It is noted that the coupling between thetwo paths in FIG. 7 is relatively small.

It can be shown that the dynamic potentially inductive part may beincluded in the sums of rational fractions where each pole is given byr_(a) ^(ij)/(s−p_(a)), following a conventional scheme of using a set ofstate variables x_(i), one variable for each pole per port. For itscorresponding time-domain equations, eight state variables areintroduced for the above transfer function (equation (5)) of a four-portcircuit 600. A generally Jordan-canonical form of realization for thissystem produces the following set of equations:

$\begin{matrix}{\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4} \\x_{5} \\x_{6} \\x_{7} \\x_{8}\end{bmatrix} = {{\begin{bmatrix}p_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\0 & p_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & p_{1} & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & p_{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & p_{2} & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & p_{2} & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 0 & p_{2} & 0 \\0 & 0 & 0 & 0 & 0 & 0 & 0 & p_{2}\end{bmatrix}\mspace{14mu}\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4} \\x_{5} \\x_{6} \\x_{7} \\x_{8}\end{bmatrix}} + {\begin{bmatrix}1000 \\0100 \\0010 \\0001 \\1000 \\0100 \\0010 \\0001\end{bmatrix}\mspace{14mu}\begin{bmatrix}v_{1} \\v_{2} \\v_{3} \\v_{4}\end{bmatrix}}}} & (6) \\{\begin{bmatrix}i_{1} \\i_{2} \\i_{3} \\i_{4}\end{bmatrix} = {\begin{bmatrix}r_{1}^{11} & r_{1}^{12} & r_{1}^{13} & r_{1}^{14} & r_{2}^{11} & r_{2}^{12} & r_{2}^{13} & r_{2}^{14} \\r_{1}^{21} & r_{1}^{22} & r_{1}^{23} & r_{1}^{24} & r_{2}^{21} & r_{2}^{22} & r_{2}^{23} & r_{2}^{24} \\r_{1}^{31} & r_{1}^{32} & r_{1}^{33} & r_{1}^{34} & r_{2}^{31} & r_{1}^{32} & r_{2}^{33} & r_{2}^{34} \\r_{1}^{41} & r_{1}^{42} & r_{1}^{43} & r_{1}^{44} & r_{2}^{41} & r_{2}^{42} & r_{2}^{43} & r_{2}^{44}\end{bmatrix}\mspace{11mu}{\quad\;{\begin{bmatrix}x_{1} \\x_{2} \\x_{3} \\x_{4} \\x_{5} \\x_{6} \\x_{7} \\x_{8}\end{bmatrix} + {\begin{bmatrix}g_{11} & 0 & 0 & 0 \\0 & g_{22} & 0 & 0 \\0 & 0 & g_{33} & 0 \\0 & 0 & 0 & g_{44}\end{bmatrix}\mspace{14mu}\left\lbrack \;\begin{matrix}v_{1} \\v_{2} \\v_{3} \\v_{4}\end{matrix} \right\rbrack}}\;}}} & (7)\end{matrix}$

The above matrix form can be rewritten into a set of equations that awetransformed into a circuit. The state part of the circuit is given byeight circuits described by the following equations:x ₁ =p ₁ x ₁ +v ₁  (8)x ₂ =p ₁ x ₂ +v ₂  (9)x ₃ =p ₁ x ₃ +v ₃  (10)x ₄ =p ₁ x ₄ +v ₄  (11)x ₅ =p ₂ x ₅ +v ₁  (12)x ₆ =p ₂ x ₆ +v ₂  (13)x ₇ =p ₂ x ₇ +v ₃  (14)x ₈ =p ₂ x ₈ +v ₄  (15)

FIG. 8 illustrates an exemplary non-physical equivalent circuit 800 thatcan be made for each of the state variables of equations (8) through(15). The values of the capacitances must be chosen such that the MNA(Modified Nodal Analysis) analysis matrix in the circuit solver used iswell conditioned.

FIG. 9 illustrates exemplary macromodel circuits 900-1 through 900-4 forthe partitioned four terminal circuit 700 of FIG. 7. As shown in FIGS. 7and 9, the exemplary macromodel circuits are separated into twoportions, A and B, in such a way that they are coupled only by waveformsources. In this manner, each portion A and B can be solved separatelyso that the interaction between them can be taken into account with thetransverse waveform relaxation process. For the A part of themacromodel, it can be shown from equation (7):i ₂ =g ₂₂ v ₂ +r ₁ ²¹ x ₁ +r ₁ ²² x ₂ +r ₁ ²³ x ₃ +r ₁ ²⁴ x ₄ +r ₂ ²¹ x₅ +r ₂ ²² x ₆ +r ₂ ²³ x ₇ +r ₂ ²⁴ x ₈  (16)where the direct (intrinsic) state variable sources for the first A part900-1 are given byi _(A2) =r ₁ ²² x ₂ +r ₁ ²³ x ₃ +r ₂ ²² x ₆ +r ₂ ²³ x ₇  (17)and the waveform sources for the first A part 900-1 arei _(WA2) =r ₁ ²¹ x ₁ +r ₁ ²⁴ x ₄ +r ₂ ²¹ x ₅ +r ₂ ²⁴ x ₈  (18)and the second subcircuit 900-2 of A isi ₃ =g ₃₃ v ₃ +r ₁ ³¹ x ₁ +r ₁ ³² x ₂ +r ₁ ³³ x ₃ +r ₁ ³⁴ x ₄ +r ₂ ³¹ x₅ +r ₂ ³² x ₆ +r ₂ ³³ x ₇ +r ₁ ³⁴ x ₈  (19)where the current sources for the second A part 900-2 are given byi _(A3) =r ₁ ³² x ₂ +r ₁ ³³ x ₃ +r ₂ ³² x ₆ +r ₂ ³³ x ₇  (20)and the waveform sources for the second A part 900-2 arei _(WA3) =r ₁ ³¹ x ₁ +r ₁ ³⁴ x ₄ +r ₂ ³¹ x ₅ +r ₂ ³⁴ x ₈  (21)

The B part of the partitioned macromodel is given from equation (7):i ₁ =g ₁₁ v ₁ +r ₁ ¹¹ x ₁ +r ₁ ¹² x ₂ +r ₁ ¹³ x ₃ +r ₁ ¹⁴ x ₄ +r ₂ ¹¹ x₅ +r ₂ ¹² x ₆ +r ₂ ¹³ x ₇ +r ₂ ¹⁴ x ₈  (22)where the direct (intrinsic) state variable sources fox the first B part900-3 are given byi _(B1) =r ₁ ¹¹ +x ₁ +r ₁ ¹⁴ x ₄ +r ₂ ¹¹ x ₅ +r ₂ ¹⁴ x ₈  (23)and the waveform sources for the first B part 900-3 arei _(WB1) =r ₁ ¹² x ₂ +r ₁ ¹³ x ₃ +r ₂ ¹² x ₆ +r ₂ ¹³ x ₇  (24)For the second subcircuit 900-4 in B:i ₄ =g ₄₄ v ₄ +r ₁ ⁴¹ x ₁ +r ₁ ⁴² x ₂ +r ₁ ⁴³ x ₃ +r ₁ ⁴⁴ x ₄ +r ₂ ⁴¹ x₅ +r ₂ ⁴² x ₆ +r ₂ ⁴³ x ₇ +r ₂ ⁴⁴ x ₈  (25)where the direct state variable sources for the second B part 900-4 aregiven byi _(B4) =r ₁ ⁴¹ +x ₂ +r ₁ ⁴⁴ x ₄ +r ₂ ⁴¹ x ₅ +r ₂ ⁴⁴ x ₈  (26)and the waveform sources for the second B part 900-4 arei _(WB4) =r ₁ ⁴² +x ₂ +r ₁ ⁴³ x ₃ +r ₂ ⁴² x ₆ +r ₂ ⁴³ x ₇  (27)

Here, the voltages and currents are real circuit variables connected tothe rest of the circuit. Each equation has been arranged such that thefirst term on the right hand side always denotes the self-conductancewhile the remaining terms of the same equation denote the couplingvoltage controlled current sources from the system for each disjointsub-circuit, where v and i denote the port node voltage and currentvector. In summary, this exemplifies the process for subdividing acomponent such that the above transverse waveform relaxation algorithmcan be applied to decouple the connectors.

FIG. 10 is a flow chart describing an exemplary implementation of acircuit preprocessor 1000. As shown in FIG. 10, the circuit preprocessor1000 initially identifies the drivers and receivers associated with eachpath during step 1010. Thereafter, the circuit preprocessor 1000performs a transverse partitioning of the transmission lines during step1020 using existing techniques. Finally, the circuit preprocessor 1000performs a transverse partitioning of the connectors and other parasiticdevices during step 1030 in accordance with the present invention.

FIG. 11 is a flow chart describing an exemplary implementation of aconnector partitioning process 1100. As shown in FIG. 11, the connectorpartitioning process 1100 initially shorts the inductors and opencircuits the capacitors in the circuit during step 1110 thereafter, theconnector partitioning process 1100 follows the path of leastconductance during step 1120 and assembles the circuit elements in onesubcircuit during step 1130.

FIG. 12 is a flow chart describing an exemplary implementation of aTransversal Waveform Relaxation process 1200 according to the presentinvention. As shown in FIG. 12, the Transversal Waveform Relaxationprocess 1200 initially identifies the circuit parts for each path duringstep 1210, such as drivers (sources), connectors, transmission lines,parasitics and receivers (terminations). The circuit preprocessor 1000and connector partitioning process 1100 are performed during step 1220.Finally, a waveform relaxation is performed during step 1230.

Thus, an algorithm is provided for the fast electrical analysis ofmultipath data transmission circuits. The algorithm includes atransversal partitioning step based on the DC current paths, and the useof waveform relaxation algorithm where the coupling between the paths isimplemented using controlled current sources. At each step of therelaxation process, the electrical state of the controlled sources isdetermined by the electrical state of the multipath circuit in theprevious state. When the coupling between the different paths are weak(the case of a working design), the relaxation process takes only fewiterations to converge.

While exemplary embodiments of the present invention have been describedwith respect to digital logic blocks, as would be apparent to oneskilled in the art, various functions may be implemented in the digitaldomain as processing steps in a software program, in hardware by circuitelements or state machines, or in combination of both software andhardware. Such software may be employed in, for example, a digitalsignal processor, micro-controller, or general-purpose computers. Suchhardware and software may be embodied within circuits implemented withinan integrated circuit.

Thus, the functions of the present invention can be embodied in the formof methods and apparatuses for practicing those methods. One or moreaspects of the present invention can be embodied in the form of programcode, for example, whether stored in a storage medium, loaded intoand/or executed by a machine, or transmitted over some transmissionmedium, wherein, when the program code is loaded into and executed by amachine, such as a computer, the machine becomes an apparatus forpracticing the invention. When implemented on a general-purposeprocessor, the program code segments combine with the processor toprovide a device that operates analogously to specific logic circuits.

System and Article of Manufacture Details

As is known in the art, the methods and apparatus discussed herein maybe distributed as an article of manufacture that itself comprises acomputer readable medium having computer readable code means embodiedthereon. The computer readable program code means is operable, inconjunction with a computer system, to carry out all or some of thesteps to perform the methods or create the apparatuses discussed herein.The computer readable medium may be a recordable medium (e.g., floppydisks, hard drives, compact disks, memory cards, semiconductor devices,chips, application specific integrated circuits (ASICs)) or may be atransmission medium (e.g., a network comprising fiber-optics, theworld-wide web, cables, or a wireless channel using time-divisionmultiple access, code-division multiple access, or other radio-frequencychannel). Any medium known or developed that can store informationsuitable for use with a computer system may be used. Thecomputer-readable code means is any mechanism for allowing a computer toread instructions and data, such as magnetic variations on a magneticmedia or height variations on the surface of a compact disk.

The computer systems and servers described herein each contain a memorythat will configure associated processors to implement the methods,steps, and functions disclosed herein. The memories could be distributedor local and the processors could be distributed or singular. Thememories could be implemented as an electrical, magnetic or opticalmemory, or any combination of these or other types of storage devices.Moreover, the term “memory” should be construed broadly enough toencompass any information able to be read from or written to an addressin the addressable space accessed by an associated processor. With thisdefinition, information on a network is still within a memory becausethe associated processor can retrieve the information from the network.

It is to be understood that the embodiments and variations shown anddescribed herein are merely illustrative of the principles of thisinvention and that various modifications may be implemented by thoseskilled in the art without departing from the scope and spirit of theinvention.

1. A method for analyzing a circuit comprising one or morenon-transmission-line connecting elements, comprising: partitioning atleast one of said non-transmission-line connecting elements in atransverse manner; identifying a plurality of subcircuits each comprisedof circuit elements from a plurality of transmission lines and one ormore non-transmission-line connecting elements in a given path; whereineach of said subcircuits is associated with a path in said circuit;performing a waveform relaxation analysis between at least two of saidsubcircuits; and repeating said step of performing said waveformrelaxation analysis using waveforms determined in a previous iterationuntil convergence has occurred.
 2. The method of claim 1, furthercomprising the step of partitioning said plurality of transmission linesin a transverse manner.
 3. The method of claim 2, further comprising thestep of representing each of said plurality of transmission lines andone or more non-transmission -line connecting elements using a circuitmacromodel.
 4. The method of claim 3, wherein said circuit macromodelseparates each path that enters and exits a given element such that itis coupled only with elements suitable for waveform relaxation analysis.5. The method of claim 1, wherein said repeating step iterates accordingto a predefined schedule.
 6. The method of claim 1, wherein saididentifying step further comprises the step of one or more of shortingone or more inductors in said circuit and opening one or more capacitorsin said circuit.
 7. The method of claim 1, wherein electrical couplingsbetween the paths of said circuit are embodied using current controlledsources.
 8. The method of claim 7, wherein at each step of said waveformrelaxation analysis, the states of said sources are updated using thepath states from a previous step.
 9. The method of claim 1, wherein saididentifying step further comprises the step of following a path of leastconductance.
 10. An apparatus for analyzing a circuit comprising one ormore non-transmission-line connecting elements, comprising: a memory;and at least one processor, coupled to the memory, operative to:partition at least one of said non-transmission-line connecting elementsin a transverse manner; identify a plurality of subcircuits eachcomprised of circuit elements from a plurality of transmission lines andone or more non-transmission-line connecting elements in a given path;wherein each of said subcircuits is associated with a path in saidcircuit; perform a waveform relaxation analysis between at least two ofsaid subcircuits; and repeat said performance of said waveformrelaxation analysis using waveforms determined in a previous iterationuntil convergence has occurred.
 11. The apparatus of claim 10, whereinsaid processor is further configured to partition said plurality oftransmission lines in a transverse manner.
 12. The apparatus of claim11, wherein said processor is further configured to represent each ofsaid plurality of transmission lines and one or morenon-transmission-line connecting elements using a circuit macromodel.13. The apparatus of claim 12, wherein said circuit macromodel separateseach path that enters and exits a given element such that it is coupledonly with elements suitable for waveform relaxation analysis.
 14. Theapparatus of claim 10, wherein said repeating of said performance ofsaid waveform relaxation analysis iterates according to a predefinedschedule.
 15. The apparatus of claim 10, wherein said processor isfurther configured to short one or more inductors in said circuit oropen one or more capacitors in said circuit.
 16. The apparatus of claim10, wherein electrical couplings between the paths of said circuit areembodied using current controlled sources.
 17. The apparatus of claim16, wherein at each step of said waveform relaxation analysis, thestates of said sources are updated using the path states from a previousstep.
 18. An article of manufacture for analyzing a circuit comprisingone or more non-transmission-line connecting elements, comprising atangible machine readable storage medium containing one or more programswhich when executed implement the steps of: partitioning at least one ofsaid non-transmission-line connecting elements in a transverse manner;identifying a plurality of subcircuits each comprised of circuitelements from a plurality of transmission lines and one or morenon-transmission-line connecting elements in a given path; wherein eachof said subcircuits is associated with a path in said circuit;performing a waveform relaxation analysis between at least two of saidsubcircuits; and repeating said step of performing said waveformrelaxation analysis using waveforms determined in a previous iterationuntil convergence has occurred.
 19. The article of manufacture of claim18, further comprising instructions for the step of partitioning saidplurality of transmission lines in a transverse manner.
 20. The articleof manufacture of claim 19, further comprising instructions for the stepof representing each of said plurality of transmission lines and one ormore non-transmission-line connecting elements using a circuitmacromodel.